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My general topology textbook introduced T$_1$-spaces as the following:

A topological space $(X,\tau)$ is said to be a T$_1$-space if:

  • $\forall x \in X$, the set $\{x\}$ is closed

They also stated the following proposition:

If $(X,\tau)$ is a topological space, then:

(1) $X$ and $\emptyset$ are closed sets

(2) The intersection of any (finite or infinite) number of closed sets is closed

(3) The union of a finite number of close sets is also closed


Now, let $A \subseteq X$. We have that $A= \bigcup_{x \in A} \{x\}$. If $A$ is finite, then $A$ is the union of a finite number of sets (3), thus every finite subset is closed. If $A$ is infinite, then this is the union of a infinite number of closed sets, hence it's not closed. Doesn't this make $\tau$ the finite-closed topology?

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  • $\begingroup$ The union of an infinite number of closed sets may or may not be closed in general. Consider the usual topology on $\mathbb R$ $\endgroup$ Jul 20 '20 at 16:07
  • $\begingroup$ The discrete topology on any set is $T_1$, and in it every set is closed. In particular, if we give $\Bbb R$ the discrete topology, then every union of closed sets is closed, no matter how many sets are involved. $\endgroup$ Jul 20 '20 at 18:38
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The mistake is at

"If $A$ is infinite, then this is the union of a infinite number of closed sets, hence it's not closed"

In general, $p\to q$ is not the same as $\sim p\to \sim q$.

There are $T_1$ space which does not have finite-closed topology. $\mathbb R$ with the standard topology is one of the example.

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Every metric space is $T_1$ and an infinite union of closed sets is not a non closed set necessarily. For example $\bigcup\limits_{x\in \mathbb{R}}\{x\}$ is closed since it is the whole space or conider any other infinite closed set, $F$, of $\mathbb{R}$ for example; then $F=\bigcup\limits_{x\in F}\{x\}$ is closed.

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